Goal: the goal of this problem is to determine when the solution of the distributional equations corresponding to localization is unstable, providing an estimate of thee mobility edge on the Bethe lattice.
Techniques: stability analysis, Laplace transforms.
Problems
In this Problem we determine for which values of parameters localization is stable, estimating the critical value of disorder where the transition to a delocalised phase occurs. Recall the results of Problem 8: the real and imaginary parts of the local self energy satisfy the self-consistent equations:
These equations admit the solution
when
, which corresponds to localization. We now determine when this solution becomes unstable.
Problem 9: an estimate of the mobility edge
- Imaginary approximation and distributional equation. We consider the equations for
and neglect the terms
in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where
. Finally, we set
and
for simplicity. Show that under these assumptions the probability density for the imaginary part,
, satisfies for
Show that the Laplace transform of this distribution,
, satisfies
- The stability analysis. We now assume to be in the localized phase, when for
the distribution
. We wish to check the stability of our assumption. This is done by controlling the tails of the distribution
for finite
.
-
For finite
, we expect that typically
, and thus
should have a peak at this scale; however, we also expect [*] some power law decay
for large
.
Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform,
for
small, with
.
- Show that the equation for
gives for
small
, and therefore this is consistent provided that there exists a
solving
Behaviour of the integral

in the case of uniformily distributed disorder, for

.
- The critical disorder. Consider now local fields
taken from a uniform distribution in
. Compute
and show that it is non monotonic, with a local minimum
in the interval
. Show that
increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when
. Show that this gives the following estimate for the critical disorder
at which the transition to delocalisation occurs:
Why the critical disorder increases with
?
- [*] - Why do we expect power law tails? Recall that in first approximation
. If
is uniformly distributed, then
.
Check out: key concepts
Linearization and stability analysis, critical disorder, mobility edge.
References
- Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)